Question: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-10y^2 - 70y - 100}{y^3 - 4y}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-10(y^2 + 7y + 10)} {y(y^2 - 4)} $ $ q = -\dfrac{10}{y} \cdot \dfrac{y^2 + 7y + 10}{y^2 - 4} $ Next factor the numerator and denominator. $ q = - \dfrac{10}{y} \cdot \dfrac{(y + 2)(y + 5)}{(y + 2)(y - 2)}$ Assuming $y \neq -2$ , we can cancel the $y + 2$ $ q = - \dfrac{10}{y} \cdot \dfrac{y + 5}{y - 2}$ Therefore: $ q = \dfrac{ -10(y + 5)}{ y(y - 2)}$, $y \neq -2$